Nelsons had been working as the City of Birmingham Symphony Orchestra since 2008.
Nelsons first led the Boston Symphony in 2011, "The New York Times" reports.
The Boston Symphony Orchestra has lingered without a leader since 2011, when James Levine resigned the position.
Nelsons said he was very honored to accept the offer, calling it one of the greatest achievements in any conductor's career.
After appearing in Boston for 8 to 10 weeks during the 2014-15 season, his first as music director, Nelsons will lead 12 weeks of programs for each subsequent year of his initial five-year contract.